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This is just an empirical formula found by Kaiser for determining the necessary filter length for a given transition width. That formula is given as Equation $(7.30)$ on page $332$:

$$M=\frac{A_s-7.95}{2.285\,\Delta\omega}+1\tag{1}$$

I think that Kaiser came up with a formula for determining the filter order (hence without the $+1$ in the equation), and the authors of your book preferred to have a formula for the filter length, so they took the original formula and added $1$ to it.

Judging from some of your previous questions, you seem to be confused when it comes to the terms filter order and filter length. For FIR filters, filter length is the number of coefficients (taps). Filter order is the (minimum) number of delay elements necessary to implement the filter. It's just like with polynomials: their order is one less than their number of coefficients. E.g., a second-order polynomial has $3$ coefficients:

$$P_2(x)=a_2x^2+a_1x+a_0\tag{2}$$

Coming back to FIR filters, you always have

$$\textrm{filter length}=\textrm{filter order}+1\tag{3}$$

This is just an empirical formula found by Kaiser for determining the necessary filter length for a given transition width. That formula is given as Equation $(7.30)$ on page $332$:

$$M=\frac{A_s-7.95}{2.285\,\Delta\omega}+1\tag{1}$$

I think that Kaiser came up with a formula for determining the filter order (hence without the $+1$ in the equation), and the authors of your book preferred to have a formula for the filter length, so they took the original formula and added $1$ to it.

Judging from some of your previous questions, you seem to be confused when it comes to the terms filter order and filter length. For FIR filters, filter length is the number of coefficients (taps). Filter order is the (minimum) number of delay elements necessary to implement the filter. It's just like with polynomials: their order is one less than their number of coefficients. E.g., a second-order polynomial has $3$ coefficients:

$$P_2(x)=a_2x^2+a_1x+a_0\tag{2}$$

Coming back to FIR filters, you always have

$$\textrm{filter length}=\textrm{filter order}+1\tag{3}$$

EDIT:

Note that formula $(1)$ above is valid for a Kaiser window, which I chose according to the title of your question. Looking a bit closer at the code, it becomes clear that in the first snippet they actually use a Hamming window, whereas in the second snippet they use a Blackman window. For these windows, there are other formulas for estimating the filter length. These formulas can be found in Table 7.1 (p. 330) of the book you refer to.

This is just an empirical formula found by Kaiser for determining the necessary filter length for a given transition width. That formula is given as Equation $(7.30)$ on page $332$:

$$M=\frac{A-7.95}{2.285\,\Delta\omega}+1\tag{1}$$

I think that Kaiser came up with a formula for determining the filter order (hence without the $+1$ in the equation), and the authors of your book preferred to have a formula for the filter length, so they took the original formula and added $1$ to it.

Judging from some of your previous questions, you seem to be confused when it comes to the terms filter order and filter length. For FIR filters, filter length is the number of coefficients (taps). Filter order is the (minimum) number of delay elements necessary to implement the filter. It's just like with polynomials: their order is one less than their number of coefficients. E.g., a second-order polynomial has $3$ coefficients:

$$P_2(x)=a_2x^2+a_1x+a_0\tag{2}$$

Coming back to FIR filters, you always have

$$\textrm{filter length}=\textrm{filter order}+1\tag{3}$$

This is just an empirical formula found by Kaiser for determining the necessary filter length for a given transition width. That formula is given as Equation $(7.30)$ on page $332$:

$$M=\frac{A_s-7.95}{2.285\,\Delta\omega}+1\tag{1}$$

I think that Kaiser came up with a formula for determining the filter order (hence without the $+1$ in the equation), and the authors of your book preferred to have a formula for the filter length, so they took the original formula and added $1$ to it.

Judging from some of your previous questions, you seem to be confused when it comes to the terms filter order and filter length. For FIR filters, filter length is the number of coefficients (taps). Filter order is the (minimum) number of delay elements necessary to implement the filter. It's just like with polynomials: their order is one less than their number of coefficients. E.g., a second-order polynomial has $3$ coefficients:

$$P_2(x)=a_2x^2+a_1x+a_0\tag{2}$$

Coming back to FIR filters, you always have

$$\textrm{filter length}=\textrm{filter order}+1\tag{3}$$

This is just an empirical formula found by Kaiser for determining the necessary filter length for a given transition width. That formula is given as Equation $(7.30)$ on page $332$:

$$M=\frac{A-7.95}{2.285\,\Delta\omega}+1\tag{1}$$

I think that Kaiser came up with a formula for determining the filter order (hence without the $+1$ in the equation), and the authors of your book preferred to have a formula for the filter length, so they took the original formula and added $1$ to it.

Judging from some of your previous questions, you seem to be confused when it comes to the terms filter order and filter length. For FIR filters, filter length is the number of coefficients (taps). Filter order is the (minimum) number of delay elements necessary to implement the filter. It's just like with polynomials: their order is one less than their number of coefficients. E.g., a second-order polynomial has $3$ coefficients:

$$P_2(x)=a_2x^2+a_1x+a_0\tag{2}$$

Coming back to FIR filters, you always have

$$\textrm{filter length}=\textrm{filter order}+1\tag{3}$$

This is just an empirical formula found by Kaiser for determining the necessary filter length for a given transition width. That formula is given as Equation $(7.30)$ on page $332$:

$$M=\frac{A-7.95}{2.285\,\Delta\omega}+1\tag{1}$$

I think that Kaiser came up with a formula for determining the filter order (hence without the $+1$ in the equation), and the authors of your book preferred to have a formula for the filter length, so they took the original formula and added $1$ to it.

Judging from some of your previous questions, you seem to be confused when it comes to the terms filter order and filter length. For FIR filters, filter length is the number of coefficients (taps). Filter order is the (minimum) number of delay elements necessary to implement the filter. It's just like with polynomials: their order is one less than their number of coefficients. E.g., a second-order polynomial has $3$ coefficients:

$$P_2(x)=a_2x^2+a_1x+a_0\tag{2}$$

Coming back to FIR filters, you always have

$$\textrm{filter length}=\textrm{filter order}+1\tag{3}$$